at the end of the skin's .ini, then it updates after 1 second as you said... Furthermore, I just found that if I click the background twice before it updates, the skin actually disappears until it updates.

This to me is strange as simply adding the container and contents somehow changes when the skin updates (I am not educated in this matter at all). And again, the skin disappearing after a double click is also strange...

at the end of the skin's .ini, then it updates after 1 second as you said...

Yes, because when you click, showing up the meters belonging to the GroupContainer meter group, you don't get them visible until the container meter isn't updated. In first code you don't encounter such issues, because there is no container.
In the second code you can get visible the revealed parts of meters if you update the container meter: LeftMouseUpAction=[!ShowMeterGroup GroupContainer][!UpdateMeter ContainerMeter][!Redraw]. Without this !UpdateMeter bang, you make visible the parts of meters revealed by the container, but since the container isn't updated, you don't see them until you get it updated. This is where the newly added !UpdateMeter bang helps.